Response of LTI Systems to Complex Exponentials

Transcription

1 3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will s ha h oupu sigal is a complx xpoial sam as h ipu, muliplid by a cosa facor ha dpds o s or z: s, whr, whr s y H s H s h d y H zz H z hz Wh h oupu sigal is a cosa ims h ipu sigal, h ipu sigal is calld a igfucio of h sysm h cosa, ampliud facor, is calld h igvalu associad wih h igfucio H s is h igvalu associad wih h igfucio Similarly, H z is h igvalu associad wih h igfucio z s If h ipu sigal is rprsd as a wighd sum of complx xpoials, h h oupu is h suprposiio of idividual rsposs For xampl, for coiuous-im sigals, s s H s s H s s s s s s a a ah s a H s for discr-im sigals, z H z z z H z z az a z ah z z ah z z

2 3 Fourir sris coiuous-im Rspos o Coiuous im Complx Expoial s h Hs s s Hs h d Cosidr a coiuous-im LI sysm wih h ui impuls rspos h Suppos h ipu sigal is x s h h oupu is y x h h x whr s hx d s hd s h d Hs s s s s Hs h( ) d h( ) d is a igfucio of h LI sysm Hs is h igvalu associad wih h igfucio

3 3 3 Fourir sris coiuous-im Cosidr a LI sysm which dlays h ipu sigal i im by 3: h 3 Suppos h ipu sigal is x cos 4 cos 7 h w ow h oupu sigal should b y x 3cos43cos73 W could hav show his via covoluio y x h xhd x 3d x3 I his xampl, w will fid h oupu sigal usig igfucios ad igvalus h igvalu associad wih h igfucio s is s s s 3 3 H s h d d Nx, rprs h ipu sigal as a wighd sum of complx xpoials j4 j4 j7 j7 x cos4cos7 = j4 j4 j7 j7 s,,, ar of h form hy ar igfucos of LI sysms Fid h oupu sigal by muliplyig associad ig valus y quals j4 j4 j7 j7 H j H j H j H j = j j4 j j4 j j7 j j7 j43 j43 j73 j73 cos43cos7 3 y x 3 idd

4 4 3 Fourir sris coiuous-im Rspos o Discr im Complx Expoial z h Hzz Cosidr a discr-im LI sysm wih h ui impuls rspos h Suppos h ipu is x z h h oupu of a LI sysm is whr z y x h h x z Hz Hz z h x hz hz hz is a igfucio of h LI sysm H z h z is h igvalu associad wih h igfucio hz H z

5 5 3 Fourir sris coiuous-im Fourir Sris Rprsaio of Coiuous im Priodic Sigals Ouli Cosidr a priodic sigal x wih priod h Fourir sris rprss x as a wighd sum of priodic igfucios wih h sam priod h Fourir sris rprsaio of x is a x a j Syhsis Equaio ar rfrrd o as h Fourir cofficis or h spcral cofficis j a x d Aalysis Equaio idicas igraio ovr ay irval of lgh h j a a is h avrag powr of h spcral compo

6 6 3 Fourir sris coiuous-im Drmiaio of Fourir Cofficis Assum w ca rprs a priodic sigal x as j x a, W wa o fid wha should a b j Muliplyig o ad igraig ovr a priod, j j x d a d j a d j Sic cos jsi, ad boh cos ad si ar priodic wih priod for, ad j d is rducd o for j x d a Fially w hav j a x d

7 7 3 Fourirr sris coiuous-im Cosidr x x is priodic wih priod S h fudamal frqucyy o ad rprs h priodic sigal as j x a whr a x j j d dd choos h irval For, a j = si For, a x j j j j jj d j j j d oig for,,

8 8 3 Fourirr sris coiuous-im Wh, 4 si si si a 4 for a No ha a is ral ad v his will happ wh x is ral ad v

9 9 3 Fourirr sris coiuous-im Spcral Compos i h Fourir Rprsaio Cosidr a coiuous-im sigal x h Fourir sriss rprsaio of x a j For, x is rfrrd o as h h spcrall compo 3 j a j is h h harmoic h off, ad is priodicc wih priod rms compo priod a a a a 3 j, a j j3, a, a j, j j j3 3 priodic wih priod is d-c s harmoic d harmoic 3rd harmoic 3 Exampl From xampl 35, a a si L As for h firs im aroud idica h firs zro crossig poi dcrass, icrass As h puls widh bcoms arrowr i h im domai, h powr sprads widr i h frqucy domai

10 3 Fourir sris coiuous-im Covrgc of h Fourir Sris Suds ar urgd o rad h scio 34 of h xboo Exisc of h Fourir Cofficis a is boudd if h sigal x is absoluly igrabl ovr a sigl priod Proof a x d a x d x d j j j x d Howvr, xisc of h Fourir cofficis dos o cssarily ma h Fourir sris rprsaio is idical o h origial priodic sigal

11 3 Fourir sris coiuous-im Approxima Rprsaio of h Sigal Approxima h priodic sigal x wih N harmoics N Dfi h rror N x a N () x x, N ad is rgy i a priod E () d N N N j W accp h followig sams wihou proof If h sigal x h x has a fii rgy ovr a sigl priod, i, d, lim E N N As N icrass, E N dcrass If x is discoiuous a, h h Fourir sris rprsaio bcoms h avrag of h valus o ihr sid of h discoiuiy

12 3 Fourirr sris coiuous-im I h xampl blow, h pa ampliud of h rippls rmais uchagd Howvr, as N icrass, h rgy i h rippls dimiishs his ffc is ow as h Gibbs phomo

13 3 3 Fourir sris coiuous-im Propris of Coiuous im Fourir Sris Sigals ar priodic wih priod, ad Liariy x y a b Ax B y c Aa Bb im Shif j x a x b a b a b x d x j j d j j x d shifig h igraio irval by

14 4 3 Fourir sris coiuous-im Exampl: Cosidr xampl 35 agai Obai x by subsiuig,, a j j si j x j x x b a x b b j j j j j j a x 5x 5x j 5 5 x b b j 3 c j 4, j v odd Fourir cofficis ar purly imagiary ad odd, for h sigal is ral ad odd x 5 5 A alraiv way is o cosidr x 3 x 5 x 3 b, j 3 c j, j j No for ay igr

15 5 3 Fourir sris coiuous-im im Rvrsal x a x b a If x is v; a is v, a a If x is odd; a is odd, a a b x d a j j x d j x d j x d j x d j x d pic h priod, l im Scalig a rmais uchagd Obsrv x a a j x a a j j x is priodic wih priod h fudamal frqucy bcoms Howvr a rmais uchags, j

16 6 3 Fourir sris coiuous-im Muliplicaio x y a b x y c a b x y x y a b j j c j j j j No a b a b hrfor c mus qual a b a b Cojuga x a x b a If x is ral, a a ad hus a a If x is ral ad v, a a a h Fourir cofficis ar ral ad v; If x is ral ad odd, a a a h Fourir cofficis ar purly imagiary ad odd b x d a x x j j j d d

17 7 3 Fourir sris coiuous-im Diffriaio x d x a x b ja d a j d d x a d d j a j j Igraio x a providd a xd b a for j Lmma j x d is priodic if ad oly if a x d x d Dfi g x d Assum x d so ha g is priodic x d g d L x a ad g b h a j b For, b a j

18 8 3 Fourir sris coiuous-im Parsval s Rlaio h avrag powr is h sum of avrag powrs i all harmoics j a d a x d a is h avrag powr of h h harmoic compo Problm 346 for proof x x d a b a a a a S o zro o hav x a x b a j x d a

19 9 3 Fourir sris coiuous-im Impuls rai x a x j j Sic x is ral ad v, a d d a is ral ad v q x ( ) x( ) x a x b j a q ja ja j j j si Sic q is ral ad odd, q is purly imagiary ad odd q g g g g j si j si is ral ad v, bcaus g d g d q for j g, j d is ral ad v